curve

As a mathematical term, "function" was coined by Gottfried Leibniz in 1694, to describe a quantity related to a curve, such as a curve's slope at a specific point of a curve. The functions Leibniz considered are today called differentiable functions, and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about limits and derivatives; both are measurements of the output or the change in the output as it depends on the input or the ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - The vocabulary of functions, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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More generally, if we have a differentiable tensor field T of rank (p,q) and a differentiable vector field Y (i.e. a differentiable section of the tangent bundle TM), then we can define the Lie derivative of T along Y. Let φ:M×R→M be the one-parameter semigroup of local diffeomorphisms of M induced by the vector flow of Y and denote φt(p) := φ(p, t). For each sufficiently small ...

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Lie derivative, Lie derivative - Definition, Lie derivative - The Lie derivative of a function, Lie derivative - The Lie derivative of a vector field, Lie derivative - The Lie derivative of differential forms, Lie derivative - Properties, Lie derivative - Lie derivative of tensor fields, Lie derivative - Coordinate expressions, Lie derivative - Generalizations, Lie derivative - Nijenhuis-Lie derivative

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The Lie derivative has a number of properties. Let be the algebra of functions defined on the manifold M. Then is a derivation on the algebra . That is, is R-linear and . Similarly, it is a derivation on where is the set of vector fields on M: which is may also be written in the equivalent notation where the tensor product symbol is used to emphasize the fact that the product of a function times a vector f ...

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Lie derivative, Lie derivative - Definition, Lie derivative - The Lie derivative of a function, Lie derivative - The Lie derivative of a vector field, Lie derivative - The Lie derivative of differential forms, Lie derivative - Properties, Lie derivative - Lie derivative of tensor fields, Lie derivative - Coordinate expressions, Lie derivative - Generalizations, Lie derivative - Nijenhuis-Lie derivative

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Let xa be a system of coordinates. For a type (r,s) tensor field T, the Lie derivative along X is here, the notation ∇ means taking the gradient in the x coordinate system. Alternatively, if we are using a torsion-free connection, then ∇ could also mean the covariant derivative. For a ...

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Lie derivative, Lie derivative - Definition, Lie derivative - The Lie derivative of a function, Lie derivative - The Lie derivative of a vector field, Lie derivative - The Lie derivative of differential forms, Lie derivative - Properties, Lie derivative - Lie derivative of tensor fields, Lie derivative - Coordinate expressions, Lie derivative - Generalizations, Lie derivative - Nijenhuis-Lie derivative

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Various generalizations of the Lie derivative play an important rôle in differential geometry. Lie derivative - Nijenhuis-Lie derivative. This article has defined the usual Lie derivative of a differential form along a vector field. One generalization, due to Albert Nijenhuis, allows one to define the Lie derivative of a differential form along any contravariant tensor field. In detail, if K is a contravariant tensor and α is a differential p-form, then it is possible define the inte ...

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Lie derivative, Lie derivative - Definition, Lie derivative - The Lie derivative of a function, Lie derivative - The Lie derivative of a vector field, Lie derivative - The Lie derivative of differential forms, Lie derivative - Properties, Lie derivative - Lie derivative of tensor fields, Lie derivative - Coordinate expressions, Lie derivative - Generalizations, Lie derivative - Nijenhuis-Lie derivative

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After an infinite number of iterations, a Menger sponge will remain. The number of cubes increases by : 20n. Where n is the number of iterations performed on the first cube: At the first level, no iterations are performed, (20 n=0 = 1). ...

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Menger sponge, Menger sponge - Construction, Menger sponge - Properties, Menger sponge - Formal definition

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The Lie derivative can also be defined on differential forms. In this context, it is closely related to the exterior derivative. Both the Lie derivative and the exterior derivative attempt to capture the idea of a derivative in different ways. These differences can be bridged by introducing the idea of an antiderivation or equivalently an interior product, after which the relationships fall out as a set of identities. Let M be a manifold and X a vector field on M. Let be a k+1-form. The interior product of See also:

Lie derivative, Lie derivative - Definition, Lie derivative - The Lie derivative of a function, Lie derivative - The Lie derivative of a vector field, Lie derivative - The Lie derivative of differential forms, Lie derivative - Properties, Lie derivative - Lie derivative of tensor fields, Lie derivative - Coordinate expressions, Lie derivative - Generalizations, Lie derivative - Nijenhuis-Lie derivative

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Newton's law of universal gravitation states the following: Every point mass attracts every other point mass by a force directed along the line connecting the two. This force is proportional to the product of the masses and inversely proportional to the square of the distance between them: where: F is the magnitude of the (repulsive) gravitational force between the two point masses G is the gravitational constant m1 is the ma ...

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Gravity, Gravity - Overview of the history of gravitational theory, Gravity - The Earth's gravity, Gravity - Comparative gravities of the Earth Sun Moon and planets, Gravity - Mathematical equations for a falling body, Gravity - Gravitational potential, Gravity - Acceleration relative to the rotating Earth, Gravity - Gravity and astronomy, Gravity - Self-gravitating system, Gravity - Practical uses of gravity, Gravity - Newton's law of universal gravitation, Gravity - Acceleration due to gravity, Gravity - Bodies with spatial extent, Gravity - Vector form, Gravity - Gravitational field, Gravity - Problems with Newton's theory, Gravity - Theoretical concerns, Gravity - Disagreement with observation, Gravity - Newton's reservations, Gravity - Einstein's theory of gravitation, Gravity - Experimental tests, Gravity - Comparison with electromagnetic force, Gravity - Gravity and quantum mechanics, Gravity - Alternative theories, Gravity - Recent alternative theories, Gravity - Historical alternative theories, Gravity - Notes

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Thee acceleration due to gravity at the Earth's surface, denoted g, is approximately 9.8 m/s2 (metres per second squared) or 32 ft/sec2. This means that, ignoring air resistance, an object falling freely near the earth's surface increases in speed by 9.8 m/s (around 22 mph) for each second of its descent. Thus, an object starting from rest will attain a speed of 9.8 m/s after one second, 19.6 m/s after two seconds, and so on. The earth itself experiences an equal and opposite force to that of the falling object, ...

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Gravity, Gravity - Overview of the history of gravitational theory, Gravity - Newton's law of universal gravitation, Gravity - Acceleration due to gravity, Gravity - Bodies with spatial extent, Gravity - Vector form, Gravity - Gravitational field, Gravity - The Earth's gravity, Gravity - Comparative gravities of the Earth Sun Moon and planets, Gravity - Mathematical equations for a falling body, Gravity - Gravitational potential, Gravity - Acceleration relative to the rotating Earth, Gravity - Gravity and astronomy, Gravity - Self-gravitating system, Gravity - Practical uses of gravity, Gravity - Problems with Newton's theory, Gravity - Theoretical concerns, Gravity - Disagreement with observation, Gravity - Newton's reservations, Gravity - Einstein's theory of gravitation, Gravity - Experimental tests, Gravity - Comparison with electromagnetic force, Gravity - Gravity and quantum mechanics, Gravity - Alternative theories, Gravity - Recent alternative theories, Gravity - Historical alternative theories, Gravity - Notes

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Indifference curves are typically assumed to have the following features: An Indifference curve slopes downward from left to right (negative slope). The negative slope is a consequence of the fact that the demand for one commodity (X) increases while the demand for another commodity (Y) decreases (because of diminishing marginal utility of Y), which is necessary to maintain the total satisfaction. Indifference curves do not intersect. This is a consequence of the assumption that the preference relation is tra ...

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Indifference curve, Indifference curve - History, Indifference curve - Preference Relations and Utility, Indifference curve - Preference Relations, Indifference curve - Formal link to Utility theory, Indifference curve - Indifference Curve Properties, Indifference curve - Indifference Map, Indifference curve - Assumptions, Indifference curve - Examples of Indifference Curves, Indifference curve - Application

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The Laffer curve and supply side economics inspired the Kemp-Roth Tax Cut of 1981. Supply-side advocates of tax cuts claimed that lower tax rates would generate more revenue because government was operating on the right-hand side of the curve. David Stockman, Reagan's budget director during his first administration and one of the early proponents of supply-side economics, maintained that the Laffer curve was not to be taken literally — at least not in the economic environment of the 1980s United States. In The Triumph of P ...

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Laffer curve, Laffer curve - Context in US History, Laffer curve - Critiques of the Laffer Curve, Laffer curve - Supporting Examples, Laffer curve - Difficulties of measurement, Laffer curve - Keynesian critique, Laffer curve - The wrong incentives?, Laffer curve - Estimates of the effectiveness of the Laffer Curve, Laffer curve - Precedents to the Laffer Curve

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The idea inherent in the Laffer curve has been described many times prior to Laffer, including: The 19th century French economist Frédéric Bastiat The 14th century Islamic scholar Ibn Khaldun The 20th century economist John Maynard Keynes The 18th century politician Alexander Hamilton The 19th century constitution of the Confederate States of America Note that Laffer himself does not claim credit for the idea [3], although he does seem to be responsible for ...

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Laffer curve, Laffer curve - Context in US History, Laffer curve - Critiques of the Laffer Curve, Laffer curve - Supporting Examples, Laffer curve - Difficulties of measurement, Laffer curve - Keynesian critique, Laffer curve - The wrong incentives?, Laffer curve - Estimates of the effectiveness of the Laffer Curve, Laffer curve - Precedents to the Laffer Curve

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Formally, a Menger sponge can be defined as follows: where M0 is the unit cube and ...

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Menger sponge, Menger sponge - Construction, Menger sponge - Properties, Menger sponge - Formal definition

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Thee acceleration due to gravity at the Earth's surface, denoted g, is approximately 9.8 m/s2 (metres per second squared) or 32 ft/sec2. This means that, ignoring air resistance, an object falling freely near the earth's surface increases in speed by 9.8 m/s (around 22 mph) for each second of its descent. Thus, an object starting from rest will attain a speed of 9.8 m/s after one second, 19.6 m/s after two seconds, and so on. The earth itself experiences an equal and opposite force to that of the falling object, ...

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Gravity, Gravity - Overview of the history of gravitational theory, Gravity - The Earth's gravity, Gravity - Comparative gravities of the Earth Sun Moon and planets, Gravity - Mathematical equations for a falling body, Gravity - Gravitational potential, Gravity - Acceleration relative to the rotating Earth, Gravity - Gravity and astronomy, Gravity - Self-gravitating system, Gravity - Practical uses of gravity, Gravity - Newton's law of universal gravitation, Gravity - Acceleration due to gravity, Gravity - Bodies with spatial extent, Gravity - Vector form, Gravity - Gravitational field, Gravity - Problems with Newton's theory, Gravity - Theoretical concerns, Gravity - Disagreement with observation, Gravity - Newton's reservations, Gravity - Einstein's theory of gravitation, Gravity - Experimental tests, Gravity - Comparison with electromagnetic force, Gravity - Gravity and quantum mechanics, Gravity - Alternative theories, Gravity - Recent alternative theories, Gravity - Historical alternative theories, Gravity - Notes

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Let us consider the rotation f around the axis u = i + j + k, with a rotation angle of 120°—i.e., 2π ⁄ 3 radians. The length of u is √3, the half angle is π ⁄ 3 (60°) with cosine ½ (cos 60° = 0.5) and sine √3 ⁄ 2 (sin 60° = 0.866). We are therefore dealing with a conjugation by the unit quaternion See also:

Quaternions and spatial rotation, Quaternions and spatial rotation - Introducion, Quaternions and spatial rotation - Non-commutativity, Quaternions and spatial rotation - Double covering, Quaternions and spatial rotation - Chirality, Quaternions and spatial rotation - Definitions, Quaternions and spatial rotation - Concepts, Quaternions and spatial rotation - Terminology, Quaternions and spatial rotation - Notation, Quaternions and spatial rotation - Reflections and Rotations, Quaternions and spatial rotation - Analytic form of a reflection, Quaternions and spatial rotation - Rotation: the composition of two reflections, Quaternions and spatial rotation - Quaternion representation of a rotation, Quaternions and spatial rotation - General rotations in four dimensional space, Quaternions and spatial rotation - Algebraic rules, Quaternions and spatial rotation - Other properties, Quaternions and spatial rotation - Quaternion rotation, Quaternions and spatial rotation - An example, Quaternions and spatial rotation - Quaternions versus other representations of rotations, Quaternions and spatial rotation - Pairs of unit quaternions as rotations in 4D space

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It is well known that the vector product is related to rotation in space. The goal then is to find a formula which expresses rotation in 3D space using quaternion multiplication, similar to the formula for a rotation in 2D using complex multiplication, f(w) = zw, where z = eα ...

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Quaternions and spatial rotation, Quaternions and spatial rotation - Introducion, Quaternions and spatial rotation - Non-commutativity, Quaternions and spatial rotation - Double covering, Quaternions and spatial rotation - Chirality, Quaternions and spatial rotation - Definitions, Quaternions and spatial rotation - Concepts, Quaternions and spatial rotation - Terminology, Quaternions and spatial rotation - Notation, Quaternions and spatial rotation - Reflections and Rotations, Quaternions and spatial rotation - Analytic form of a reflection, Quaternions and spatial rotation - Rotation: the composition of two reflections, Quaternions and spatial rotation - Quaternion representation of a rotation, Quaternions and spatial rotation - General rotations in four dimensional space, Quaternions and spatial rotation - Algebraic rules, Quaternions and spatial rotation - Other properties, Quaternions and spatial rotation - Quaternion rotation, Quaternions and spatial rotation - An example, Quaternions and spatial rotation - Quaternions versus other representations of rotations, Quaternions and spatial rotation - Pairs of unit quaternions as rotations in 4D space

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The representation of a rotation as a quaternion (4 numbers) is more compact than the representation as an orthogonal matrix (9 numbers). Furthermore, for a given axis and angle, one can easily construct the corresponding quaternion, and conversely, for a given quaternion one can easily read off the axis and the angle. Both of these are much harder with matrices or Euler angles. In computer games and other applications, one is often interested in “smooth rotations,” meaning that the scene should slowly rotate and not in a single s ...

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Quaternions and spatial rotation, Quaternions and spatial rotation - Introducion, Quaternions and spatial rotation - Non-commutativity, Quaternions and spatial rotation - Double covering, Quaternions and spatial rotation - Chirality, Quaternions and spatial rotation - Definitions, Quaternions and spatial rotation - Concepts, Quaternions and spatial rotation - Terminology, Quaternions and spatial rotation - Notation, Quaternions and spatial rotation - Reflections and Rotations, Quaternions and spatial rotation - Analytic form of a reflection, Quaternions and spatial rotation - Rotation: the composition of two reflections, Quaternions and spatial rotation - Quaternion representation of a rotation, Quaternions and spatial rotation - General rotations in four dimensional space, Quaternions and spatial rotation - Algebraic rules, Quaternions and spatial rotation - Other properties, Quaternions and spatial rotation - Quaternion rotation, Quaternions and spatial rotation - An example, Quaternions and spatial rotation - Quaternions versus other representations of rotations, Quaternions and spatial rotation - Pairs of unit quaternions as rotations in 4D space

Read more here: » Quaternions and spatial rotation: Encyclopedia II - Quaternions and spatial rotation - Quaternions versus other representations of rotations

A pair of unit quaternions zl and zr can represent any rotation in 4D space. Given a four dimensional vector v, and pretending that it is a quaternion, we can rotate the vector v like this: It is straightforward to check that for each matrix M MT = I, that is, that each matrix (and hence both matrices together) represents a rotation. Note that since (zl v) zr = zl ( ...

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Quaternions and spatial rotation, Quaternions and spatial rotation - Introducion, Quaternions and spatial rotation - Non-commutativity, Quaternions and spatial rotation - Double covering, Quaternions and spatial rotation - Chirality, Quaternions and spatial rotation - Definitions, Quaternions and spatial rotation - Concepts, Quaternions and spatial rotation - Terminology, Quaternions and spatial rotation - Notation, Quaternions and spatial rotation - Reflections and Rotations, Quaternions and spatial rotation - Analytic form of a reflection, Quaternions and spatial rotation - Rotation: the composition of two reflections, Quaternions and spatial rotation - Quaternion representation of a rotation, Quaternions and spatial rotation - General rotations in four dimensional space, Quaternions and spatial rotation - Algebraic rules, Quaternions and spatial rotation - Other properties, Quaternions and spatial rotation - Quaternion rotation, Quaternions and spatial rotation - An example, Quaternions and spatial rotation - Quaternions versus other representations of rotations, Quaternions and spatial rotation - Pairs of unit quaternions as rotations in 4D space

Read more here: » Quaternions and spatial rotation: Encyclopedia II - Quaternions and spatial rotation - Pairs of unit quaternions as rotations in 4D space

In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible "directions" in which one can pass through p. The elements of the tangent space are called tangent vectors at p. All the tangent spaces have the same dimension, equal to the dimension of the manifold. For example, if the given manifold is a 2-sphere, one can picture the tangent space at a point as the plane which touches the sphere at that ...

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Tangent space, Tangent space - Informal description, Tangent space - Formal definitions, Tangent space - Definition as directions of curves, Tangent space - Definition via derivations, Tangent space - Definition via the cotangent space, Tangent space - Properties, Tangent space - Tangent vectors as directional derivatives, Tangent space - The derivative of a map

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Consider the quaternions with modulus 1. They form a multiplicative group, acting on R3: for any such quaternion , the mapping f(x) = z x z* is a counterclockwise rotation through an angle about an axis v; −z is the same rotation. Composition of arbitrary rotations in R3 corresponds to the fairly simple operation of quaternion multiplication. A pair of quaternions also allows for compact representations of rotations in 4D s ...

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Quaternions and spatial rotation, Quaternions and spatial rotation - Introducion, Quaternions and spatial rotation - Non-commutativity, Quaternions and spatial rotation - Double covering, Quaternions and spatial rotation - Chirality, Quaternions and spatial rotation - Definitions, Quaternions and spatial rotation - Concepts, Quaternions and spatial rotation - Terminology, Quaternions and spatial rotation - Notation, Quaternions and spatial rotation - Reflections and Rotations, Quaternions and spatial rotation - Analytic form of a reflection, Quaternions and spatial rotation - Rotation: the composition of two reflections, Quaternions and spatial rotation - Quaternion representation of a rotation, Quaternions and spatial rotation - General rotations in four dimensional space, Quaternions and spatial rotation - Algebraic rules, Quaternions and spatial rotation - Other properties, Quaternions and spatial rotation - Quaternion rotation, Quaternions and spatial rotation - An example, Quaternions and spatial rotation - Quaternions versus other representations of rotations, Quaternions and spatial rotation - Pairs of unit quaternions as rotations in 4D space

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Every quaternion z = a + bi + cj + dk can be viewed as a sum a + u of a real number a (called the “real part” of the quaternion) and a 3-vector u = (b, c, d) = bi + cj + dk in R3 (called the “imaginary part”). Two such quaternions are added by adding the real parts and the imaginary parts separately: (a + u) + (bSee also:

Quaternions and spatial rotation, Quaternions and spatial rotation - Introducion, Quaternions and spatial rotation - Non-commutativity, Quaternions and spatial rotation - Double covering, Quaternions and spatial rotation - Chirality, Quaternions and spatial rotation - Definitions, Quaternions and spatial rotation - Concepts, Quaternions and spatial rotation - Terminology, Quaternions and spatial rotation - Notation, Quaternions and spatial rotation - Reflections and Rotations, Quaternions and spatial rotation - Analytic form of a reflection, Quaternions and spatial rotation - Rotation: the composition of two reflections, Quaternions and spatial rotation - Quaternion representation of a rotation, Quaternions and spatial rotation - General rotations in four dimensional space, Quaternions and spatial rotation - Algebraic rules, Quaternions and spatial rotation - Other properties, Quaternions and spatial rotation - Quaternion rotation, Quaternions and spatial rotation - An example, Quaternions and spatial rotation - Quaternions versus other representations of rotations, Quaternions and spatial rotation - Pairs of unit quaternions as rotations in 4D space

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Einstein's theory of gravitation answered the problems with Newton's theory noted above. In a revolutionary move, his theory of general relativity (1915) stated that the presence of mass, energy, and momentum causes spacetime to become curved. Because of this curvature, the paths that objects in inertial motion follow can "deviate" or change direction over time. This deviation appears to us as an acceleration towards massive objects, which Newton characterized as being gravity. In general relativity however, this acceleration or free fall is ...

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Gravity, Gravity - Overview of the history of gravitational theory, Gravity - The Earth's gravity, Gravity - Comparative gravities of the Earth Sun Moon and planets, Gravity - Mathematical equations for a falling body, Gravity - Gravitational potential, Gravity - Acceleration relative to the rotating Earth, Gravity - Gravity and astronomy, Gravity - Self-gravitating system, Gravity - Practical uses of gravity, Gravity - Newton's law of universal gravitation, Gravity - Acceleration due to gravity, Gravity - Bodies with spatial extent, Gravity - Vector form, Gravity - Gravitational field, Gravity - Problems with Newton's theory, Gravity - Theoretical concerns, Gravity - Disagreement with observation, Gravity - Newton's reservations, Gravity - Einstein's theory of gravitation, Gravity - Experimental tests, Gravity - Comparison with electromagnetic force, Gravity - Gravity and quantum mechanics, Gravity - Alternative theories, Gravity - Recent alternative theories, Gravity - Historical alternative theories, Gravity - Notes

Read more here: » Gravity: Encyclopedia II - Gravity - Einstein's theory of gravitation